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The average weight of a certain number o...

The average weight of a certain number of students in a group is 54 kg. if 12 students of average weight 52 kg join the group, then the average weight of all the students in the group decreases by 750 g. what was the numberof students, initially in the group?

A

18

B

24

C

15

D

20

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem step by step, let's define the variables and follow the logic of the question. ### Step 1: Define Variables Let \( x \) be the initial number of students in the group. ### Step 2: Calculate Initial Total Weight The average weight of the initial group is 54 kg. Therefore, the total weight of the initial group can be calculated as: \[ \text{Total weight of initial group} = x \times 54 \] ### Step 3: Calculate Total Weight After New Students Join When 12 students with an average weight of 52 kg join the group, their total weight is: \[ \text{Total weight of new students} = 12 \times 52 = 624 \text{ kg} \] ### Step 4: Calculate New Total Weight The new total weight of the group after the new students join is: \[ \text{New total weight} = (x \times 54) + 624 \] ### Step 5: Calculate New Total Number of Students The new total number of students in the group is: \[ \text{New total number of students} = x + 12 \] ### Step 6: Calculate New Average Weight According to the problem, the average weight decreases by 750 grams (which is 0.75 kg). Therefore, the new average weight is: \[ \text{New average weight} = 54 - 0.75 = 53.25 \text{ kg} \] ### Step 7: Set Up the Equation Using the new average weight, we can set up the equation: \[ \frac{(x \times 54) + 624}{x + 12} = 53.25 \] ### Step 8: Cross-Multiply to Eliminate the Fraction Cross-multiplying gives us: \[ (x \times 54) + 624 = 53.25 \times (x + 12) \] ### Step 9: Expand the Right Side Expanding the right side: \[ (x \times 54) + 624 = 53.25x + 639 \] ### Step 10: Rearrange the Equation Rearranging the equation to isolate \( x \): \[ 54x + 624 - 53.25x - 639 = 0 \] \[ 0.75x - 15 = 0 \] ### Step 11: Solve for \( x \) Adding 15 to both sides: \[ 0.75x = 15 \] Dividing both sides by 0.75: \[ x = \frac{15}{0.75} = 20 \] ### Conclusion The initial number of students in the group was \( \boxed{20} \).
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